"Mirrorless, mirrorless, on the wall, who's the fairest camera of all?"
But seriously.... When it comes to sports shooting, there are a number of reasons mirrorless don't cut it.
1. By they time you hang a 70-200/2.8, 300mm or 400mm lens on the front of it, the small size and lighter weight of a mirrorless camera is more a liability, than a benefit. The diminutive camera doesn't balance well with the bigger lenses that sports shooting pros are likely to be using. it's made even worse if you also need to use an adapter, since there aren't many 70-200/2.8, 300/2.8 or 400/2.8 being made for mirrorless yet.
2. For the speed and action of sports, an optical viewfinder and a separate phase detection sensor array are still superior to an electronic viewfinder and dual pixel phase detection embedded in the imaging sensor. Even worse would be a mirrorless without any viewfinder... ever tried to shoot sports using Live View and holding the same at arms length... in bright sunlight? Rots of ruck with that, Scooby!
3. Limited lens selection (already touched on that above), especially of the types that sports photogs need and use... Nikon and Canon each have around 90 lenses in their systems (roughly 1/3 are crop only in each). Pentax and Sony... around 40 (Sony is actually less because some are A-mount, some are E-mount). Few MILC systems have more than 25 or 30 to choose among tops, and some have only 10 or fewer. Also, due to their current popularity, they also tend to be more expensive, both the MILC itself and the lenses to designed for use upon it, when compared to the most comparable DSLR gear.
For example, Sony is seeking to compete with pro-grade cameras that are up to the task.... bBut so far has only three lenses in E-mount that are remotely "sports oriented": two 70-200s and a 100-400mm. Where are the 300/2.8, 400/2.8, 500/4 and 600/4 we see on the sidelines at the Super Bowl? Okay, Sony's got 300/2.8 and 500/4... in A-mount, that can be used on their MILC with an adapter. In comparison.... for pro sports shooters Canon's got four 70-200s, 100-400, two 300mm, three 400mm, 500mm, 600mm, 800mm, plus a 200-400 with a built-in/matched 1.4X. Nikon has a similar range to choose among (some differences: 80-400 & 200-500, instead of 100-400... only two 70-200s... and their 200-400 doesn't have a built in TC). Fourteen or fifteen lenses with Nikon or Canon... versus five in the Sony line (two of which require adapters to be used on their MILC).
4. Also, tiny cameras mean tiny batteries with very limited shots-per-charge... and there are few accessories such as battery grips to help the situation. I use a grip on my DLSRs that allow me to get around 2500 shots with a pair of batteries. And some day's shooting an 8 or 10 hour event I shoot more than that and need to swap batteries once. Many MILC only get 600, 500, 400 or even 300 shots per charge. It takes "juice" to run those electronic viewfinders and LCD screens (the screen on my DSLR is turned off... I only see image reviews on demand, to save power). With many MILC I'd be changing batteries 4 to 6 times in a day's shoot! (P.S. I looked for info about shots per charge for the Sony batteries, but can't find any.... Both Canon and Nikon list CIPA standard battery test results for their cameras, which I can beat by a large margin with some simple power saving tricks.)
5. Battery grips also provide a secondary set of controls for vertical orientation. If you shoot a lot of vertical/portrait oriented shots, it's a lot more comfortable than continually "twisting" your wrist to get the shot. I'd estimate about half my shots are horiz., half vert. Someone shooting for a print magazine might take a lot more verticals. There simply aren't many vertical/battery grips available for MILC yet. Fujifilm makes a really nice one... for one of their camera models. Sony makes a nice one too... for their full frame camera. It costs $50 to $100 more than the most expensive Canon battery grips. As far as I can tell, Sony doesn't make one for their APS-C camera (the format I prefer to shoot sports), but Vello does? (3rd party... lower cost... but build quality? reliability? durability?)
6. In general, with a few exceptions, MILC are simply more amateur to advanced amateur-oriented... so far. Manufacturers have treated them more as a "step up" from point n shoot, but below mid-grade, pro-grade DSLRs. More automation, less direct access to the camera controls that pros tend to want and use. Canon sure did with their M-series, but appears to have begun changing that. So did Nikon and Pentax with their essentially failed MILC lines. They were probably all paranoid about eroding their own DSLR sales (which happened anyway, just to other brands instead.) Sony, Fuji, Oly and Panasonic all treated at least some their MILC models more seriously. Still, especially on the smaller MILC, there's limited "real estate"... not a lot of room for all the buttons and dials that a pro-grade camera typically uses. I've heard and read more than one MILC user complain about too small controls that are easy to miss or mistake. A sports shooter is often making adjustments by touch and familiarity with the controls, while keeping their eye to the viewfinder. That would be hard to do with some MILC.
7. As new people move into the field of sports photography and the manufacturers make more pro-oriented models and expand their MILC-specific lens lines, I'm sure we'll see more folks using them now that they offer AF performance and other features that are necessary. But seasoned pros aren't quick to jump brands (unless someone else is paying for their gear... an employer or a sponsor). They have a lot of money invested... not to mention the learning curve of any new system. Pro sports shooters often have to react instantly and the ability to do that only comes with a lot of familiarity with thei gear. Switching brands means "starting over", in a lot of ways. And it can be very expensive!
Besides, we're stubborn curmudgeons who don't like all that new fangled stuff!
Actually, I think MILC are cool and really want one... for some things (street photography, travel, vintage manual focus lenses). But SPORTS is NOT one of them. I'll keep my DSLRs for that. The MILC I'll get will be more for fun... so it's not gotten high priority.